The Israeli military said four other soldiers were wounded in the raid by fighters who had used a concealed tunnel to reach Israel. One Hamas gunman was killed, while the rest of the group managed to escape back into the Gaza Strip, the military added.
Israel sent ground forces into Hamas-controlled Gaza on Thursday after 10 days of air and naval barrages failed to stop rocket fire from the Palestinian territory.
It has vowed to destroy the tunnel network and hunt down the militants' stockpiles of missiles.
The land incursion has so far failed to subdue Hamas and its allies, who fired more than 90 rockets into Israel on Saturday, the Israeli army said. One killed a man and wounded two children in a southern Bedouin Arab village in Israel, police said.
Gaza officials said at least 335 Palestinians, including 70 children, have been killed in the 12-day conflict - about 100 in the last 48 hours alone. On Israel's side, three soldiers and two civilians have died.
Diplomatic efforts to secure a ceasefire involving, amongst others, Egypt, Qatar, France and the United Nations, have failed to make headway. Hamas rejected an initial attempt by Cairo last week to end the fighting.
The sound of explosions rocked Gaza through the day, while sirens repeatedly sounded in towns and cities in southern and central Israel, warning of approaching missiles.
Eight members of a Palestinian family, including a six-month old baby and two children aged three and 13, died overnight when an Israeli shell hit their house in Beit Hanoun, in northeastern Gaza, officials said.
"Oh, God. Oh, God," a shocked relative said, repeatedly knocking his head against a wall as rescuers searched for survivors.
Hostilities between the two sides escalated following the killing last month of three Jewish students that Israel blames on Hamas. Hamas neither confirmed nor denied involvement. The apparent revenge murder of a Palestinian youth in Jerusalem, for which Israel has charged three Jews, further fuelled tensions.
Israeli bulldozers and engineers worked along a mile-wide strip of Gaza's eastern frontier, uncovering 13 tunnels, at least one of them 30 metres (90 feet) deep, military spokesman Lieutenant-Colonel Peter Lerner said.
About 95 rocket launchers were also found and destroyed in the sweep, he added.
Searches were continuing on what he described as an open-ended mission that had "severely impeded Hamas capabilities".
Israel says more than 1,500 rockets have been fired out of Gaza during this month's fighting, and between 3,000-4,000 others destroyed in military strikes - together almost half of the militants' original estimated arsenal.
Hamas says it is continuously replenishing its stock of weapons and is ready for a prolonged conflict.
The Israeli death toll has been kept low due to the rockets' relative inaccuracy, a network of air raid sirens and shelters and the Iron Dome rocket interceptor's 90 percent success rate.
The Israeli military urged Palestinians to flee an ever-growing area of Gaza ahead of further military action in the Mediterranean enclave. Locals say about half of the territory's 1.8-million population have been told to move.
With both the Israeli and Egyptian borders sealed off, Gazans say they have few places to escape to.
The largest United Nations agency in Gaza, UNRWA, said about 61,500 people had sought refugee in its buildings, mainly schools - more than in any previous conflict there between Israel and Islamist militants.
"My family and I left the house with nothing but the clothes we were wearing at two in the morning," said Kareem Ramadan, 45, an unemployed father of three, resting in an UNWRA school in northern Gaza.
"Maybe we'll go back and see the house destroyed, but at least we're alive."
U.N. Secretary-General Ban Ki-moon plans to travel to the region this weekend as part of a growing international efforts to end hostilities, while French Foreign Minister Laurent Fabius met officials in both Egypt and Jordan to discuss the crisis.
"Our absolute priority is to have a ceasefire. Egypt has put forward a good initiative," Fabius said in Jordan, before heading to Israel for more talks.
However, Egypt said on Saturday it had no plans to revise the ceasefire proposal, which Hamas has already rejected.
Pro-Palestinian protesters took to the streets of numerous European cities, including London and Paris, to demand an immediate end to the conflict.
"There will be no truce without an end to the war that the Occupation (Israel) began, a lifting of the blockade and a halt to all violations and killings in Gaza and the West Bank," Hamas spokesman Sami Abu Zuhri said in Gaza.